What does the number of required expert demonstrations in Imitation Learning depend on?
The answer to your question can be found in the original paper that introduced the max-margin and projection imitation learning (IL) algorithms: Apprenticeship Learning via Inverse Reinforcement Learning (by Abbel and Ng, 2004, ICML). Specifically, theorem 1 (section 4, page 4) states

Let an $\text{MDP} \setminus R$, features $ \phi : S \rightarrow [0, 1]^k$, and any $\epsilon &gt; 0$ be given. Then the apprenticeship learning algorithm (both max-margin and projection versions) will terminate with $t^{(i)} \leq \epsilon$ after at most
$$n=O\left(\frac{k}{(1-\gamma)^{2} \epsilon^{2}} \log \frac{k}{(1-\gamma) \epsilon}\right)$$
iterations.

Here $k$ is the dimension of the feature vectors, so it's clear that the number of iterations needed for these algorithms to terminate scales with $k$. The proof of this theorem can be found in appendix A of the same paper (and all other terms are defined in the paper, which you should read to understand all the details). Of course, this result holds (only) for these specific IL algorithms (which are the algorithms the author of your slides, Abbel, is referring to). See also theorem 2 and the experiments section (in particular, figure 4, which shows the performance as a function of the number of trajectories) of the same paper.
These slides provide a nice overview of the contents of this paper, so I suggest that you read them too.